Chapter 3. Derivatives
Section 3.1 Derivatives
Definition. The derivative of a function f at a number a, denoted by f ′ (a), is
f (a + h) − f (a)
f (x) − f (a)
= lim
x→a
h→0
h
x−a
f ′ (a) = lim
if the limit exist.
Example 1. Find f ′ (a) if f (x) =
√
2x − 3, a > 3/2.
Geometric interpretation of the derivative. f ′ (a) is the slope of the tangent line to
y = f (x) at the point (a, f (a)).
√
Example 2. Find an equation of the tangent line to f (x) = 2x − 3 at the point (2,1).
Other interpretations of the derivative.
• f ′ (a) is the instanteneous rate of change of y = f (x) with respect to x when x = a.
• if s = f (t) is the position function of a particle that moves along a straight line, then
f ′ (a) is the velocity of the particle at time t = a
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A function
f (x + h) − f (x)
h→0
h
f ′ (x) = lim
is called the derivative of f .
Definition. A function f is differentiable at a if f ′ (a) exists. It is differentiable on an
open interval (a, b) if it is differentiable at every number in the interval.
Example 3. Where is the function f (x) = |x − 2| differentiable?
Theorem. If f is differentiable at a, then f is continuous at a
When is the function not differentiable at x = a?
• f has a ”corner” or ”kink” at a
• f is discontinuous at a
• the curve y = f (x) has a vertical tangent line at x = a
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